Distributive Equations of Fuzzy Implications Based on Continuous Triangular Conorms Given as Ordinal Sums

Aifang Xie,Huawen Liu,Cheng Li

doi:10.1109/tfuzz.2012.2221719

Abstract

Recently, the distributive equations of fuzzy implications based on t -norms or t-conorms have become a focus of research. The solutions to these equations can help people design the structures of fuzzy systems in such a way that the number of rules is largely reduced. This paper studies the distributive functional equation I(x,S 1 ( y , z ))= S 2 ( I ( x , y ), I ( x , z )), where S 1 and S 2 are two continuous t -conorms given as ordinal sums, and I :[0,1] 2 → [0,1] is a binary function which is increasing with respect to the second place. If there is no summand of S 2 in the interval [ I (1,0), I (1,1)], we get its continuous solutions directly. If there are summands of S 2 in the interval [ I (1,0), I (1,1)], by defining a new concept called feasible correspondence and using this concept, we describe the solvability of the distributive equation above and characterize its general continuous solutions. When I is restricted to fuzzy implications, it is showed that there is no continuous solution to this equation. We characterize its fuzzy implication solutions, which are continuous on (0,1] × [0,1].

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